Due to the categorical interpretation of the stack \(\mathfrak {PT}_{\mathfrak {X}/\mathfrak {C}}^P\)  (resp. \(\mathfrak {PT}_{\mathfrak {Y}/\mathfrak {A}}^{P}\) ) as in Section 5.2, the representatives of its objects we will consider are \(S_{\xi }\) -flat families of stable orbifold PT pairs \(\varphi : \mathcal {O}_{\mathtt {X}_{S_{\xi }}}\rightarrow \mathcal {F}\)  on \(\mathtt {X}_{S_{\xi }}\)  (resp. \(\varphi : \mathcal {O}_{\mathtt {Y}_{S_{\xi }}}\rightarrow \mathcal {F}\)  on \(\mathtt {Y}_{S_{\xi }}\) ) with fixed Hilbert homomorphism \(P: K^0(\mathcal {X})\rightarrow \mathbb {Z}\)  (resp. \(P: K^0(\mathcal {Y})\rightarrow \mathbb {Z}\) ). Since \(\mathcal {F}\)  is an \(S_{\xi }\) -flat family of pure 1-dimensional sheaves on \(\mathtt {X}_{S_{\xi }}\)  (resp. \(\mathtt {Y}_{S_{\xi }}\) ),
the associated generalized Hilbert polynomial should be a polynomial of degree one, i.e., \(P(H^{\otimes m})=am+b\)  with \(a,b\in \mathbb {Z}\)  and \(a>0\)  where \(H=\pi ^*\mathcal {O}_{X}(1)\)  (resp. \(H=\pi ^*\mathcal {O}_{Y}(1)\) ). The boundedness here means that two stacks \(\mathfrak {PT}_{\mathfrak {X}/\mathfrak {C}}^{P}\)  and \(\mathfrak {PT}_{\mathfrak {Y}/\mathfrak {A}}^{P}\)  are of finite type. By [[1]}, p. 100] and Theorem REF , we only need to show that these two stacks are quasi-compact.
